{"id":230,"date":"2011-12-03T00:28:16","date_gmt":"2011-12-02T15:28:16","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=230"},"modified":"2021-10-03T20:25:50","modified_gmt":"2021-10-03T11:25:50","slug":"kbr200902","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr200902\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2009\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(xyz\\) \u7a7a\u9593\u5185\u306b\u304a\u3044\u3066, \\(yz\\) \u5e73\u9762\u4e0a\u3067\u653e\u7269\u7dda \\(z = y^2\\) \u3068\u76f4\u7dda \\(z = 4\\) \u3067\u56f2\u307e\u308c\u308b\u5e73\u9762\u56f3\u5f62\u3092 \\(D\\) \u3068\u3059\u308b.\r\n\u70b9 \\(( 1, 1, 0 )\\) \u3092\u901a\u308a \\(z\\) \u8ef8\u306b\u5e73\u884c\u306a\u76f4\u7dda\u3092 \\(\\ell\\) \u3068\u3057, \\(\\ell\\) \u306e\u307e\u308f\u308a\u306b \\(D\\) \u3092 \\(1\\) \u56de\u8ee2\u3055\u305b\u3066\u3067\u304d\u308b\u7acb\u4f53\u3092 \\(E\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(D\\) \u3068\u5e73\u9762 \\(z=t\\) \u3068\u306e\u4ea4\u308f\u308a\u3092 \\(D _ t\\) \u3068\u3059\u308b. \u305f\u3060\u3057 \\(0 \\leqq t \\leqq 4\\) \u3068\u3059\u308b. \u70b9P\u304c \\(D _ t\\) \u4e0a\u3092\u52d5\u304f\u3068\u304d, \u70b9P\u3068\u70b9 \\(( 1, 1, t )\\) \u3068\u306e\u8ddd\u96e2\u306e\u6700\u5927\u5024, \u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u5e73\u9762 \\(z=t\\) \u306b\u3088\u308b \\(E\\) \u306e\u5207\u308a\u53e3\u306e\u9762\u7a4d \\(S(t) \\ ( 0 \\leqq t \\leqq 4 )\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(E\\) \u306e\u4f53\u7a4d \\(V\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\[\r\nD _ t : \\ x = 0 , \\ z = t \\quad \\left( -\\sqrt{t} \\leqq y \\leqq \\sqrt{t} \\right)\r\n\\]\r\n\u306a\u306e\u3067, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(0 \\leqq t \\leqq 1\\) \u306e\u3068\u304d\r\n\"\"\r\n\u6700\u5927\u5024 \\(R _ t\\) \u306f\r\n\\[\r\nR _ t = \\sqrt{\\left( 1+\\sqrt{t} \\right)^2 +1} = \\sqrt{t +2\\sqrt{t} +2}\r\n\\]\r\n\u6700\u5c0f\u5024 \\(r _ t\\) \u306f\r\n\\[\r\nr _ t = \\sqrt{\\left( 1-\\sqrt{t} \\right)^2 +1} = \\sqrt{t -2\\sqrt{t} +2}\r\n\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(1 \\lt t \\leqq 4\\) \u306e\u3068\u304d\r\n\"\"\r\n\u6700\u5927\u5024 \\(R _ t\\) \u306f, 1*<\/strong> \u306e\u3068\u304d\u3068\u540c\u69d8\u306b\r\n\\[\r\nR _ t = \\sqrt{t +2\\sqrt{t} +2}\r\n\\]\r\n\u6700\u5c0f\u5024 \\(r _ t\\) \u306f\r\n\\[\r\nr _ t = 1\r\n\\]<\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u6700\u5927\u5024\u306f\r\n\\[\r\nR _ t = \\underline{\\sqrt{t +2\\sqrt{t} +2}}\r\n\\]\r\n\u6700\u5c0f\u5024\u306f\r\n\\[\r\nr _ t =\\underline{\\left\\{ \\begin{array}{ll} \\sqrt{t -2\\sqrt{t} +2} & \\ \\left( \\ 0 \\leqq t \\leqq 1 \\text{\u306e\u3068\u304d} \\right) \\\\ 1 & \\ \\left( \\ 0 \\leqq t \\leqq 4 \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\r\n\\]\r\n

      (2)<\/strong><\/p>\r\n

      \\[\r\nS(t) = \\pi {R _ t}^2 -\\pi {r _ t}^2 = \\pi \\left( {R _ t}^2 -{r _ t}^2 \\right)\r\n\\]\r\n\u306a\u306e\u3067, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nS(t) = \\underline{\\left\\{ \\begin{array}{ll} 4 \\pi \\sqrt{t} & \\ \\left( \\ 0 \\leqq t \\leqq 1 \\text{\u306e\u3068\u304d} \\right) \\\\ \\pi \\left( \\sqrt{t}+1 \\right)^2 & \\ \\left( \\ 0 \\leqq t \\leqq 4 \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\r\n\\]\r\n

      (3)<\/strong><\/p>\r\n

      \\[\\begin{align}\r\nV & = \\displaystyle\\int _ 0^4 S(t) \\, dt \\\\\r\n& = 4 \\pi \\displaystyle\\int _ 0^1 \\sqrt{t} \\, dt +\\pi \\displaystyle\\int _ 1^4 \\left( t +2\\sqrt{t}+1 \\right) \\, dt \\\\\r\n& = 4 \\pi \\left[ \\dfrac{2 t^{\\frac{3}{2}}}{3} \\right] _ 0^1 +\\pi \\left[ \\dfrac{t^2}{2} +\\dfrac{4 t^{\\frac{3}{2}}}{3} +t \\right] _ 1^4 \\\\\r\n& = \\dfrac{8 \\pi}{3} +\\pi \\left( 8 +\\dfrac{32}{3} +4 \\right) -\\pi \\left( \\dfrac{1}{2} +\\dfrac{4}{3} +1 \\right) \\\\\r\n& = \\underline{\\dfrac{45 \\pi}{2}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xyz\\) \u7a7a\u9593\u5185\u306b\u304a\u3044\u3066, \\(yz\\) \u5e73\u9762\u4e0a\u3067\u653e\u7269\u7dda \\(z = y^2\\) \u3068\u76f4\u7dda \\(z = 4\\) \u3067\u56f2\u307e\u308c\u308b\u5e73\u9762\u56f3\u5f62\u3092 \\(D\\) \u3068\u3059\u308b. \u70b9 \\(( 1, 1, 0 )\\) \u3092\u901a\u308a \\(z\\) \u8ef8\u306b … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[39],"tags":[144,15],"class_list":["post-230","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2009","tag-tsukuba_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/230","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=230"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/230\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=230"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=230"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=230"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}